Noncollapsibility of the odds ratio
Noncollapsibility
Assume having a binary outcome \(Y\) and a treatment indicator \(X\) denoting if patient was randomised to treatment (\(X=1\)) or control (\(X=0\)). The analysis would be to fit a simple logistic regression model
\[ P(Y=1|X)=\text{expit}(\alpha_0+\alpha_1X), \]
with \(\text{expit}(\alpha_0+\alpha_1X)=\frac{\exp(\alpha_0+\alpha_1X)}{1+\exp(\alpha_0+\alpha_1X)}\).
Often, we write the model with the linear predictor, that is, the logit transformed expectation,
\[ \boxed{\text{logit}P(Y=1|X)=\alpha_0+{\color{red}\alpha_1}X}. \]
Of course, there will be other factors that influence the probability that \(Y=1\). Assume another variable \(C\) which influences the probability that \(Y=1\). The conditional model would be
\[ \boxed{\text{logit}P(Y=1|X,C)=\beta_0+{\color{red}\beta_1}X+\beta_2C}. \]
Now, under randomisation, \(C\) and \(X\) are (population) independent – that is – there is no confounding of the effect of \(X\) on \(Y\) by \(C\) in place.
Though \(C\) is not a confounder, in general, \({\color{red}\alpha_1 \neq \beta_1}\). The parameters (log odds ratios) in the marginal and conditional model are different. The proof is based on William’s Tower rule for conditional expectations:
\[ \begin{align} E(Y|X)=&E(E(Y|X,C)|X)\\ =&E(\text{expit}(\beta_0+\beta_1X+\beta_2C)|X)\\ \neq &\text{expit}(\beta_0+\beta_1X+\beta_2E(C|X)) \end{align} \]
With other link functions (in log-linear models) and identity (in linear models)), however, the two quantities are equal.
Example
We simulate data from logistic GLM with non-confounding covariate \(C\):
psych::headTail(dat)| C | X | Ydich | |
|---|---|---|---|
| 1 | -7.87 | 0 | 0 |
| 2 | -7.87 | 0 | 0 |
| 3 | -7.86 | 0 | 0 |
| 4 | -7.81 | 0 | 0 |
| … | … | … | … |
| 397 | 7.93 | 1 | 1 |
| 398 | 7.94 | 0 | 1 |
| 399 | 7.95 | 0 | 1 |
| 400 | 7.95 | 0 | 1 |
Marginal model
equatiomatic::extract_eq(modM)\[ \log\left[ \frac { P( \operatorname{Ydich} = \operatorname{1} ) }{ 1 - P( \operatorname{Ydich} = \operatorname{1} ) } \right] = \alpha + \beta_{1}(\operatorname{X}) \]
Conditional model
equatiomatic::extract_eq(modC)\[ \log\left[ \frac { P( \operatorname{Ydich} = \operatorname{1} ) }{ 1 - P( \operatorname{Ydich} = \operatorname{1} ) } \right] = \alpha + \beta_{1}(\operatorname{X}) + \beta_{2}(\operatorname{C}) \]
Illustrate non-collapsibility
pred<-predict(modM,type="response",se.fit=TRUE)
predgA<-predict(modC,newdata=data.frame(C=C[X==0],X=0),se.fit=TRUE,type="response")
predgB<-predict(modC,newdata=data.frame(C=C[X==1],X=1),se.fit=TRUE,type="response")
plot(C,Ydich,col=c("blue","red")[X+1],ylab="Probability")
lines(sort(C[X==0]),sort(predgA$fit),col="blue")
lines(sort(C[X==1]),sort(predgB$fit),col="red")
abline(a=mean(pii[X==0]),b=0,col="blue",lty=2)
abline(a=mean(pii[X==1]),b=0,col="red",lty=2)Marginal model
summary(modM)$coef Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.08004 0.1415 -0.5655 0.571710
X 0.63377 0.2040 3.1071 0.001889
Conditional model
Effect of \(X\) is larger, even though \(C\) is not a confounder. This is due to non-collapsibility of the odds ratio
summary(modC)$coef Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.05395 0.3208 -0.1682 8.665e-01
X 2.46270 0.5209 4.7280 2.267e-06
C 1.04913 0.1200 8.7453 2.223e-18
IPW-model
Estimate inverse probability weights to fit marginal structural models in a point treatment situation.
Weights
Marginal structural model
Marginal structural model for the causal effect of \(X\) on \(Y\) corrected for confounding by \(C\) using inverse probability weighting
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0683 0.1419 -0.4814 0.630474
X 0.6117 0.2043 2.9938 0.002927
Loading required package: survey
Loading required package: grid
Loading required package: Matrix
Loading required package: survival
Attaching package: 'survey'
The following object is masked from 'package:graphics':
dotchart
Warning in eval(family$initialize): non-integer #successes in a binomial glm!
summary(msm)
Call:
svyglm(formula = Ydich ~ X, design = svydesign(~1, weights = ~sw,
data = dat), family = "binomial")
Survey design:
svydesign(~1, weights = ~sw, data = dat)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0683 0.1417 -0.48 0.6301
X 0.6117 0.2042 3.00 0.0029
(Dispersion parameter for binomial family taken to be 1.003)
Number of Fisher Scoring iterations: 4
Comparison
modelsummary::modelsummary(models=list("marginal"=modM,"conditional"=modC,"IPW"=modW))| marginal | conditional | IPW | |
|---|---|---|---|
| (Intercept) | -0.080 | -0.054 | -0.068 |
| (0.142) | (0.321) | (0.142) | |
| X | 0.634 | 2.463 | 0.612 |
| (0.204) | (0.521) | (0.204) | |
| C | 1.049 | ||
| (0.120) | |||
| Num.Obs. | 400 | 400 | 400 |
| AIC | 543.4 | 152.6 | |
| BIC | 551.4 | 164.5 | |
| Log.Lik. | -269.718 | -73.275 | |
| F | 9.654 | 38.281 | 8.963 |
| RMSE | 0.49 | 0.23 | 0.49 |
- conditional-marginal=noncollaps+confouding
- IPW-marginal=confounding
- conditional-IPW=noncollaps
Smoking and children weight
str(bwt)'data.frame': 189 obs. of 9 variables:
$ low : int 0 0 0 0 0 0 0 0 0 0 ...
$ age : int 19 33 20 21 18 21 22 17 29 26 ...
$ lwt : int 182 155 105 108 107 124 118 103 123 113 ...
$ smoke : int 0 0 1 1 1 0 0 0 1 1 ...
$ ht : int 0 0 0 0 0 0 0 0 0 0 ...
$ ui : int 1 0 0 1 1 0 0 0 0 0 ...
$ ftv : int 0 3 1 2 0 0 1 1 1 0 ...
$ bwt : int 2523 2551 2557 2594 2600 2622 2637 2637 2663 2665 ...
$ socclass: Factor w/ 3 levels "I","II","III": 2 3 1 1 1 3 1 3 1 1 ...
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.0871 0.2147 -5.062 4.142e-07
smoke 0.7041 0.3196 2.203 2.762e-02
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.841 0.3529 -5.216 1.828e-07
smoke 1.116 0.3692 3.023 2.507e-03
socclassII 1.084 0.4900 2.212 2.693e-02
socclassIII 1.109 0.4003 2.769 5.618e-03
temp <- ipwpoint(exposure=smoke,family = "binomial",link = "logit",data=bwt,
numerator=~1,denominator=~socclass)
summary(temp$ipw.weights) Min. 1st Qu. Median Mean 3rd Qu. Max.
0.723 0.723 0.741 1.000 1.328 2.186
bwt$sw<-temp$ipw.weights Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.277 0.2270 -5.625 6.696e-08
smoke 0.937 0.3283 2.854 4.798e-03
- conditional-marginal=noncollaps+confouding
- IPW-marginal=confounding
- conditional-IPW=noncollaps
modelsummary::modelsummary(models=list("marginal"=glm_smoke,"conditional"=glm_multi_1,"IPW"=modW))| marginal | conditional | IPW | |
|---|---|---|---|
| (Intercept) | -1.087 | -1.841 | -1.277 |
| (0.215) | (0.353) | (0.227) | |
| smoke | 0.704 | 1.116 | 0.937 |
| (0.320) | (0.369) | (0.328) | |
| socclassII | 1.084 | ||
| (0.490) | |||
| socclassIII | 1.109 | ||
| (0.400) | |||
| Num.Obs. | 189 | 189 | 189 |
| AIC | 233.8 | 228.0 | |
| BIC | 240.3 | 240.9 | |
| Log.Lik. | -114.902 | -109.987 | |
| F | 4.852 | 4.294 | 8.148 |
| RMSE | 0.46 | 0.45 | 0.46 |
The view of potential outcomes
Simulate some data
library(simstudy)
# define the data
defB <- defData(varname = "C", formula =0.27,
dist = "binary")
defB <- defData(defB, varname = "Y0", formula = "-2.5 + 1.75*C",
dist = "binary", link = "logit")
defB <- defData(defB, varname = "Y1", formula = "-1.5 + 1.75*C",
dist = "binary", link = "logit")
defB <- defData(defB, varname = "X", formula = "0.315 + 0.352 *C",
dist = "binary")
defB <- defData(defB, varname = "Y", formula = "Y0 + X * (Y1 - Y0)",
dist = "nonrandom")
defB| varname | formula | variance | dist | link |
|---|---|---|---|---|
| C | 0.27 | 0 | binary | identity |
| Y0 | -2.5 + 1.75*C | 0 | binary | logit |
| Y1 | -1.5 + 1.75*C | 0 | binary | logit |
| X | 0.315 + 0.352 *C | 0 | binary | identity |
| Y | Y0 + X * (Y1 - Y0) | 0 | nonrandom | identity |
| id | C | Y0 | Y1 | X | Y |
|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 1 | 0 | 0 |
| 3 | 1 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 1 | 1 | 0 | 1 |
| 6 | 1 | 0 | 0 | 1 | 0 |
| 7 | 1 | 0 | 0 | 1 | 0 |
| 8 | 1 | 0 | 1 | 0 | 0 |
| 9 | 1 | 1 | 1 | 1 | 1 |
| 10 | 1 | 0 | 0 | 1 | 0 |
| 11 | 0 | 0 | 1 | 1 | 1 |
| 12 | 0 | 0 | 0 | 1 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 1 | 0 | 0 |
| 15 | 0 | 0 | 1 | 1 | 1 |
| 16 | 0 | 0 | 0 | 1 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 1 | 0 | 0 |
| 19 | 0 | 0 | 0 | 0 | 0 |
| 20 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 0 | 0 | 0 | 0 |
| 22 | 1 | 1 | 0 | 1 | 0 |
| 23 | 0 | 0 | 0 | 0 | 0 |
| 24 | 0 | 0 | 1 | 0 | 0 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 26 | 0 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 1 | 0 |
| 28 | 0 | 1 | 1 | 0 | 1 |
| 29 | 0 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 31 | 1 | 0 | 1 | 1 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 33 | 1 | 1 | 1 | 0 | 1 |
| 34 | 0 | 0 | 0 | 0 | 0 |
| 35 | 0 | 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 | 1 | 0 |
| 37 | 0 | 0 | 0 | 1 | 0 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 39 | 0 | 0 | 0 | 1 | 0 |
| 40 | 1 | 1 | 0 | 1 | 0 |
| 41 | 1 | 0 | 0 | 0 | 0 |
| 42 | 1 | 1 | 0 | 1 | 0 |
| 43 | 0 | 0 | 0 | 0 | 0 |
| 44 | 1 | 1 | 1 | 1 | 1 |
| 45 | 0 | 0 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 | 1 | 0 |
| 47 | 0 | 0 | 0 | 1 | 0 |
| 48 | 0 | 0 | 0 | 0 | 0 |
| 49 | 0 | 0 | 1 | 0 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 51 | 0 | 0 | 0 | 0 | 0 |
| 52 | 0 | 0 | 0 | 1 | 0 |
| 53 | 0 | 0 | 0 | 0 | 0 |
| 54 | 0 | 0 | 0 | 1 | 0 |
| 55 | 0 | 0 | 0 | 1 | 0 |
| 56 | 0 | 0 | 0 | 0 | 0 |
| 57 | 0 | 0 | 0 | 0 | 0 |
| 58 | 1 | 0 | 1 | 0 | 0 |
| 59 | 0 | 0 | 0 | 1 | 0 |
| 60 | 1 | 0 | 1 | 1 | 1 |
| 61 | 0 | 0 | 0 | 0 | 0 |
| 62 | 1 | 1 | 1 | 1 | 1 |
| 63 | 0 | 0 | 1 | 1 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 65 | 1 | 0 | 1 | 1 | 1 |
| 66 | 1 | 0 | 0 | 1 | 0 |
| 67 | 0 | 0 | 1 | 0 | 0 |
| 68 | 1 | 1 | 0 | 0 | 1 |
| 69 | 0 | 0 | 0 | 0 | 0 |
| 70 | 0 | 0 | 0 | 0 | 0 |
| 71 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 73 | 0 | 0 | 0 | 0 | 0 |
| 74 | 1 | 1 | 1 | 1 | 1 |
| 75 | 0 | 0 | 0 | 0 | 0 |
| 76 | 0 | 0 | 0 | 0 | 0 |
| 77 | 0 | 0 | 0 | 0 | 0 |
| 78 | 0 | 0 | 1 | 1 | 1 |
| 79 | 0 | 0 | 1 | 0 | 0 |
| 80 | 0 | 0 | 0 | 0 | 0 |
| 81 | 0 | 0 | 1 | 0 | 0 |
| 82 | 0 | 0 | 0 | 0 | 0 |
| 83 | 0 | 0 | 0 | 0 | 0 |
| 84 | 1 | 0 | 1 | 1 | 1 |
| 85 | 0 | 0 | 1 | 0 | 0 |
| 86 | 1 | 0 | 1 | 1 | 1 |
| 87 | 0 | 0 | 1 | 0 | 0 |
| 88 | 0 | 0 | 1 | 0 | 0 |
| 89 | 0 | 0 | 1 | 0 | 0 |
| 90 | 0 | 0 | 0 | 0 | 0 |
| 91 | 0 | 0 | 0 | 0 | 0 |
| 92 | 1 | 0 | 0 | 1 | 0 |
| 93 | 1 | 0 | 1 | 0 | 0 |
| 94 | 0 | 0 | 0 | 0 | 0 |
| 95 | 1 | 0 | 1 | 0 | 0 |
| 96 | 0 | 1 | 0 | 1 | 0 |
| 97 | 1 | 0 | 1 | 1 | 1 |
| 98 | 0 | 0 | 0 | 1 | 0 |
| 99 | 0 | 0 | 0 | 0 | 0 |
| 100 | 0 | 0 | 0 | 0 | 0 |
| 101 | 0 | 0 | 0 | 0 | 0 |
| 102 | 0 | 0 | 0 | 1 | 0 |
| 103 | 1 | 0 | 0 | 1 | 0 |
| 104 | 1 | 1 | 1 | 1 | 1 |
| 105 | 0 | 0 | 0 | 0 | 0 |
| 106 | 0 | 0 | 0 | 0 | 0 |
| 107 | 0 | 0 | 0 | 0 | 0 |
| 108 | 0 | 1 | 0 | 1 | 0 |
| 109 | 0 | 0 | 0 | 0 | 0 |
| 110 | 0 | 1 | 0 | 0 | 1 |
| 111 | 0 | 0 | 1 | 0 | 0 |
| 112 | 1 | 0 | 0 | 1 | 0 |
| 113 | 0 | 0 | 0 | 1 | 0 |
| 114 | 0 | 0 | 0 | 0 | 0 |
| 115 | 0 | 0 | 0 | 0 | 0 |
| 116 | 0 | 0 | 0 | 0 | 0 |
| 117 | 0 | 0 | 0 | 1 | 0 |
| 118 | 0 | 0 | 1 | 0 | 0 |
| 119 | 0 | 0 | 0 | 1 | 0 |
| 120 | 0 | 0 | 0 | 0 | 0 |
| 121 | 1 | 0 | 1 | 1 | 1 |
| 122 | 1 | 0 | 0 | 1 | 0 |
| 123 | 0 | 0 | 0 | 0 | 0 |
| 124 | 0 | 0 | 0 | 0 | 0 |
| 125 | 0 | 0 | 1 | 0 | 0 |
| 126 | 1 | 0 | 1 | 0 | 0 |
| 127 | 1 | 1 | 0 | 1 | 0 |
| 128 | 0 | 0 | 0 | 0 | 0 |
| 129 | 1 | 0 | 1 | 1 | 1 |
| 130 | 0 | 0 | 0 | 1 | 0 |
| 131 | 0 | 0 | 1 | 1 | 1 |
| 132 | 1 | 0 | 0 | 1 | 0 |
| 133 | 1 | 1 | 0 | 1 | 0 |
| 134 | 0 | 0 | 0 | 0 | 0 |
| 135 | 0 | 0 | 1 | 1 | 1 |
| 136 | 0 | 0 | 0 | 0 | 0 |
| 137 | 0 | 0 | 0 | 0 | 0 |
| 138 | 0 | 0 | 0 | 0 | 0 |
| 139 | 0 | 1 | 0 | 0 | 1 |
| 140 | 0 | 0 | 0 | 0 | 0 |
| 141 | 0 | 0 | 0 | 1 | 0 |
| 142 | 1 | 0 | 1 | 1 | 1 |
| 143 | 0 | 0 | 0 | 0 | 0 |
| 144 | 1 | 0 | 0 | 0 | 0 |
| 145 | 0 | 0 | 0 | 1 | 0 |
| 146 | 0 | 0 | 0 | 1 | 0 |
| 147 | 0 | 0 | 0 | 1 | 0 |
| 148 | 0 | 0 | 0 | 0 | 0 |
| 149 | 0 | 0 | 0 | 0 | 0 |
| 150 | 0 | 0 | 0 | 0 | 0 |
| 151 | 0 | 0 | 0 | 0 | 0 |
| 152 | 0 | 0 | 1 | 0 | 0 |
| 153 | 1 | 0 | 1 | 1 | 1 |
| 154 | 0 | 0 | 0 | 0 | 0 |
| 155 | 0 | 0 | 1 | 0 | 0 |
| 156 | 1 | 1 | 0 | 1 | 0 |
| 157 | 0 | 1 | 0 | 0 | 1 |
| 158 | 0 | 0 | 0 | 0 | 0 |
| 159 | 0 | 0 | 0 | 0 | 0 |
| 160 | 0 | 0 | 1 | 0 | 0 |
| 161 | 0 | 0 | 1 | 0 | 0 |
| 162 | 1 | 0 | 0 | 0 | 0 |
| 163 | 0 | 0 | 1 | 0 | 0 |
| 164 | 1 | 0 | 0 | 1 | 0 |
| 165 | 0 | 0 | 0 | 0 | 0 |
| 166 | 0 | 0 | 0 | 0 | 0 |
| 167 | 0 | 0 | 0 | 0 | 0 |
| 168 | 0 | 0 | 0 | 0 | 0 |
| 169 | 1 | 1 | 0 | 1 | 0 |
| 170 | 0 | 0 | 1 | 1 | 1 |
| 171 | 0 | 0 | 0 | 0 | 0 |
| 172 | 0 | 0 | 0 | 1 | 0 |
| 173 | 1 | 1 | 0 | 1 | 0 |
| 174 | 1 | 0 | 1 | 1 | 1 |
| 175 | 0 | 0 | 0 | 0 | 0 |
| 176 | 0 | 0 | 0 | 0 | 0 |
| 177 | 0 | 0 | 0 | 1 | 0 |
| 178 | 1 | 0 | 0 | 1 | 0 |
| 179 | 0 | 0 | 0 | 1 | 0 |
| 180 | 1 | 1 | 0 | 1 | 0 |
| 181 | 1 | 1 | 1 | 0 | 1 |
| 182 | 0 | 0 | 0 | 1 | 0 |
| 183 | 0 | 0 | 0 | 1 | 0 |
| 184 | 1 | 0 | 1 | 0 | 0 |
| 185 | 0 | 0 | 0 | 1 | 0 |
| 186 | 0 | 0 | 0 | 0 | 0 |
| 187 | 0 | 0 | 0 | 1 | 0 |
| 188 | 0 | 0 | 0 | 1 | 0 |
| 189 | 0 | 0 | 0 | 0 | 0 |
| 190 | 0 | 0 | 0 | 0 | 0 |
| 191 | 0 | 0 | 0 | 0 | 0 |
| 192 | 0 | 0 | 0 | 0 | 0 |
| 193 | 0 | 0 | 0 | 0 | 0 |
| 194 | 0 | 0 | 0 | 0 | 0 |
| 195 | 0 | 0 | 0 | 1 | 0 |
| 196 | 1 | 0 | 1 | 1 | 1 |
| 197 | 0 | 0 | 0 | 0 | 0 |
| 198 | 1 | 0 | 1 | 1 | 1 |
| 199 | 1 | 0 | 0 | 1 | 0 |
| 200 | 0 | 0 | 0 | 1 | 0 |
| 201 | 0 | 1 | 0 | 1 | 0 |
| 202 | 1 | 0 | 0 | 0 | 0 |
| 203 | 1 | 1 | 1 | 0 | 1 |
| 204 | 0 | 0 | 0 | 0 | 0 |
| 205 | 0 | 0 | 0 | 0 | 0 |
| 206 | 0 | 0 | 0 | 0 | 0 |
| 207 | 0 | 0 | 1 | 0 | 0 |
| 208 | 1 | 0 | 1 | 0 | 0 |
| 209 | 0 | 0 | 1 | 1 | 1 |
| 210 | 1 | 0 | 0 | 1 | 0 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 212 | 0 | 0 | 0 | 1 | 0 |
| 213 | 0 | 0 | 0 | 0 | 0 |
| 214 | 1 | 1 | 0 | 1 | 0 |
| 215 | 0 | 0 | 1 | 1 | 1 |
| 216 | 0 | 0 | 0 | 0 | 0 |
| 217 | 1 | 0 | 0 | 1 | 0 |
| 218 | 0 | 0 | 0 | 1 | 0 |
| 219 | 0 | 0 | 0 | 0 | 0 |
| 220 | 0 | 0 | 0 | 1 | 0 |
| 221 | 0 | 0 | 0 | 0 | 0 |
| 222 | 0 | 0 | 0 | 0 | 0 |
| 223 | 1 | 1 | 0 | 1 | 0 |
| 224 | 0 | 0 | 0 | 0 | 0 |
| 225 | 1 | 0 | 1 | 0 | 0 |
| 226 | 0 | 0 | 0 | 0 | 0 |
| 227 | 0 | 0 | 0 | 1 | 0 |
| 228 | 1 | 0 | 0 | 0 | 0 |
| 229 | 0 | 0 | 0 | 0 | 0 |
| 230 | 1 | 0 | 1 | 1 | 1 |
| 231 | 0 | 0 | 0 | 1 | 0 |
| 232 | 0 | 0 | 0 | 0 | 0 |
| 233 | 0 | 0 | 0 | 0 | 0 |
| 234 | 1 | 0 | 1 | 1 | 1 |
| 235 | 0 | 0 | 1 | 0 | 0 |
| 236 | 1 | 0 | 1 | 0 | 0 |
| 237 | 0 | 0 | 0 | 0 | 0 |
| 238 | 1 | 1 | 0 | 1 | 0 |
| 239 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 241 | 0 | 0 | 0 | 0 | 0 |
| 242 | 0 | 0 | 1 | 0 | 0 |
| 243 | 0 | 0 | 0 | 0 | 0 |
| 244 | 0 | 0 | 0 | 0 | 0 |
| 245 | 0 | 0 | 0 | 0 | 0 |
| 246 | 1 | 0 | 1 | 1 | 1 |
| 247 | 1 | 0 | 1 | 0 | 0 |
| 248 | 1 | 0 | 0 | 0 | 0 |
| 249 | 0 | 0 | 0 | 0 | 0 |
| 250 | 0 | 0 | 0 | 0 | 0 |
| 251 | 1 | 0 | 0 | 1 | 0 |
| 252 | 0 | 0 | 0 | 1 | 0 |
| 253 | 0 | 0 | 0 | 0 | 0 |
| 254 | 0 | 0 | 0 | 0 | 0 |
| 255 | 0 | 0 | 0 | 1 | 0 |
| 256 | 0 | 0 | 0 | 0 | 0 |
| 257 | 0 | 0 | 0 | 0 | 0 |
| 258 | 1 | 1 | 1 | 0 | 1 |
| 259 | 0 | 0 | 0 | 0 | 0 |
| 260 | 0 | 0 | 0 | 0 | 0 |
| 261 | 0 | 0 | 0 | 0 | 0 |
| 262 | 0 | 0 | 0 | 0 | 0 |
| 263 | 0 | 0 | 0 | 1 | 0 |
| 264 | 0 | 0 | 1 | 1 | 1 |
| 265 | 0 | 0 | 0 | 0 | 0 |
| 266 | 1 | 0 | 0 | 1 | 0 |
| 267 | 0 | 0 | 0 | 0 | 0 |
| 268 | 0 | 0 | 0 | 0 | 0 |
| 269 | 0 | 0 | 0 | 0 | 0 |
| 270 | 0 | 0 | 0 | 0 | 0 |
| 271 | 0 | 0 | 1 | 1 | 1 |
| 272 | 0 | 0 | 0 | 0 | 0 |
| 273 | 0 | 0 | 0 | 0 | 0 |
| 274 | 1 | 0 | 1 | 0 | 0 |
| 275 | 1 | 0 | 1 | 0 | 0 |
| 276 | 0 | 0 | 0 | 0 | 0 |
| 277 | 0 | 0 | 0 | 1 | 0 |
| 278 | 0 | 0 | 0 | 0 | 0 |
| 279 | 0 | 0 | 0 | 1 | 0 |
| 280 | 0 | 0 | 0 | 1 | 0 |
| 281 | 1 | 1 | 1 | 0 | 1 |
| 282 | 0 | 0 | 0 | 0 | 0 |
| 283 | 0 | 0 | 0 | 0 | 0 |
| 284 | 1 | 0 | 1 | 0 | 0 |
| 285 | 1 | 1 | 1 | 0 | 1 |
| 286 | 0 | 0 | 0 | 0 | 0 |
| 287 | 0 | 0 | 0 | 0 | 0 |
| 288 | 1 | 0 | 1 | 0 | 0 |
| 289 | 0 | 0 | 0 | 1 | 0 |
| 290 | 0 | 0 | 0 | 0 | 0 |
| 291 | 0 | 0 | 1 | 0 | 0 |
| 292 | 0 | 0 | 0 | 1 | 0 |
| 293 | 1 | 0 | 1 | 1 | 1 |
| 294 | 0 | 0 | 0 | 0 | 0 |
| 295 | 1 | 0 | 1 | 1 | 1 |
| 296 | 0 | 0 | 0 | 1 | 0 |
| 297 | 0 | 0 | 0 | 1 | 0 |
| 298 | 1 | 1 | 1 | 1 | 1 |
| 299 | 0 | 0 | 0 | 1 | 0 |
| 300 | 0 | 0 | 1 | 0 | 0 |
| 301 | 0 | 0 | 1 | 1 | 1 |
| 302 | 1 | 0 | 1 | 1 | 1 |
| 303 | 0 | 0 | 0 | 0 | 0 |
| 304 | 0 | 0 | 0 | 0 | 0 |
| 305 | 1 | 0 | 0 | 1 | 0 |
| 306 | 0 | 0 | 1 | 0 | 0 |
| 307 | 0 | 0 | 1 | 0 | 0 |
| 308 | 1 | 0 | 0 | 1 | 0 |
| 309 | 1 | 0 | 1 | 1 | 1 |
| 310 | 0 | 0 | 0 | 1 | 0 |
| 311 | 0 | 1 | 0 | 0 | 1 |
| 312 | 0 | 0 | 1 | 0 | 0 |
| 313 | 0 | 0 | 0 | 0 | 0 |
| 314 | 0 | 0 | 0 | 1 | 0 |
| 315 | 0 | 0 | 0 | 1 | 0 |
| 316 | 0 | 0 | 0 | 0 | 0 |
| 317 | 1 | 0 | 1 | 0 | 0 |
| 318 | 0 | 1 | 0 | 0 | 1 |
| 319 | 1 | 0 | 1 | 1 | 1 |
| 320 | 1 | 1 | 1 | 1 | 1 |
| 321 | 0 | 0 | 0 | 0 | 0 |
| 322 | 0 | 0 | 1 | 1 | 1 |
| 323 | 0 | 0 | 0 | 0 | 0 |
| 324 | 0 | 0 | 0 | 0 | 0 |
| 325 | 1 | 0 | 1 | 1 | 1 |
| 326 | 0 | 0 | 0 | 0 | 0 |
| 327 | 0 | 0 | 0 | 1 | 0 |
| 328 | 0 | 0 | 0 | 1 | 0 |
| 329 | 0 | 0 | 0 | 0 | 0 |
| 330 | 0 | 0 | 0 | 0 | 0 |
| 331 | 0 | 0 | 0 | 0 | 0 |
| 332 | 0 | 0 | 1 | 0 | 0 |
| 333 | 0 | 0 | 0 | 0 | 0 |
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| 862 | 0 | 1 | 0 | 0 | 1 |
| 863 | 1 | 0 | 1 | 1 | 1 |
| 864 | 0 | 0 | 0 | 1 | 0 |
| 865 | 0 | 0 | 0 | 0 | 0 |
| 866 | 0 | 0 | 1 | 0 | 0 |
| 867 | 0 | 1 | 0 | 0 | 1 |
| 868 | 1 | 0 | 1 | 1 | 1 |
| 869 | 1 | 0 | 1 | 0 | 0 |
| 870 | 0 | 0 | 0 | 0 | 0 |
| 871 | 1 | 0 | 1 | 0 | 0 |
| 872 | 0 | 0 | 1 | 0 | 0 |
| 873 | 0 | 0 | 0 | 0 | 0 |
| 874 | 0 | 0 | 0 | 0 | 0 |
| 875 | 0 | 0 | 0 | 0 | 0 |
| 876 | 0 | 0 | 0 | 0 | 0 |
| 877 | 1 | 0 | 0 | 0 | 0 |
| 878 | 0 | 1 | 1 | 0 | 1 |
| 879 | 0 | 0 | 0 | 0 | 0 |
| 880 | 1 | 0 | 1 | 0 | 0 |
| 881 | 0 | 0 | 0 | 0 | 0 |
| 882 | 0 | 0 | 0 | 0 | 0 |
| 883 | 0 | 0 | 0 | 0 | 0 |
| 884 | 0 | 0 | 0 | 1 | 0 |
| 885 | 1 | 0 | 1 | 1 | 1 |
| 886 | 0 | 0 | 0 | 1 | 0 |
| 887 | 1 | 0 | 0 | 1 | 0 |
| 888 | 1 | 0 | 1 | 1 | 1 |
| 889 | 0 | 0 | 0 | 0 | 0 |
| 890 | 1 | 1 | 0 | 0 | 1 |
| 891 | 1 | 0 | 1 | 1 | 1 |
| 892 | 0 | 0 | 1 | 0 | 0 |
| 893 | 1 | 0 | 1 | 0 | 0 |
| 894 | 0 | 0 | 0 | 0 | 0 |
| 895 | 0 | 0 | 0 | 0 | 0 |
| 896 | 1 | 1 | 0 | 1 | 0 |
| 897 | 0 | 0 | 0 | 0 | 0 |
| 898 | 0 | 0 | 0 | 0 | 0 |
| 899 | 0 | 0 | 0 | 0 | 0 |
| 900 | 0 | 0 | 0 | 1 | 0 |
| 901 | 1 | 0 | 0 | 1 | 0 |
| 902 | 1 | 0 | 0 | 1 | 0 |
| 903 | 1 | 1 | 0 | 1 | 0 |
| 904 | 1 | 1 | 0 | 0 | 1 |
| 905 | 0 | 0 | 1 | 1 | 1 |
| 906 | 1 | 0 | 1 | 1 | 1 |
| 907 | 0 | 0 | 0 | 0 | 0 |
| 908 | 0 | 0 | 1 | 0 | 0 |
| 909 | 0 | 0 | 1 | 0 | 0 |
| 910 | 0 | 0 | 0 | 0 | 0 |
| 911 | 0 | 0 | 1 | 1 | 1 |
| 912 | 1 | 1 | 1 | 1 | 1 |
| 913 | 0 | 0 | 1 | 0 | 0 |
| 914 | 1 | 0 | 1 | 1 | 1 |
| 915 | 0 | 0 | 0 | 1 | 0 |
| 916 | 1 | 0 | 1 | 0 | 0 |
| 917 | 0 | 0 | 0 | 0 | 0 |
| 918 | 0 | 0 | 0 | 1 | 0 |
| 919 | 0 | 0 | 0 | 1 | 0 |
| 920 | 0 | 0 | 0 | 0 | 0 |
| 921 | 0 | 0 | 0 | 0 | 0 |
| 922 | 0 | 0 | 0 | 0 | 0 |
| 923 | 0 | 0 | 0 | 0 | 0 |
| 924 | 0 | 0 | 0 | 1 | 0 |
| 925 | 0 | 0 | 0 | 1 | 0 |
| 926 | 0 | 0 | 1 | 0 | 0 |
| 927 | 0 | 0 | 0 | 0 | 0 |
| 928 | 1 | 0 | 0 | 0 | 0 |
| 929 | 0 | 0 | 0 | 0 | 0 |
| 930 | 0 | 0 | 0 | 0 | 0 |
| 931 | 1 | 0 | 0 | 1 | 0 |
| 932 | 0 | 0 | 0 | 0 | 0 |
| 933 | 1 | 0 | 1 | 1 | 1 |
| 934 | 0 | 0 | 0 | 1 | 0 |
| 935 | 0 | 0 | 1 | 1 | 1 |
| 936 | 1 | 1 | 1 | 1 | 1 |
| 937 | 0 | 0 | 0 | 0 | 0 |
| 938 | 0 | 1 | 1 | 1 | 1 |
| 939 | 0 | 0 | 0 | 1 | 0 |
| 940 | 0 | 0 | 0 | 1 | 0 |
| 941 | 0 | 0 | 0 | 0 | 0 |
| 942 | 0 | 0 | 0 | 0 | 0 |
| 943 | 1 | 0 | 1 | 0 | 0 |
| 944 | 1 | 1 | 1 | 1 | 1 |
| 945 | 0 | 1 | 0 | 1 | 0 |
| 946 | 0 | 0 | 1 | 1 | 1 |
| 947 | 1 | 0 | 1 | 1 | 1 |
| 948 | 0 | 0 | 0 | 0 | 0 |
| 949 | 0 | 0 | 0 | 0 | 0 |
| 950 | 1 | 0 | 1 | 1 | 1 |
| 951 | 1 | 1 | 0 | 1 | 0 |
| 952 | 0 | 0 | 0 | 0 | 0 |
| 953 | 1 | 0 | 1 | 0 | 0 |
| 954 | 1 | 1 | 0 | 0 | 1 |
| 955 | 0 | 0 | 1 | 1 | 1 |
| 956 | 0 | 0 | 0 | 0 | 0 |
| 957 | 0 | 0 | 0 | 0 | 0 |
| 958 | 0 | 0 | 1 | 1 | 1 |
| 959 | 0 | 0 | 0 | 0 | 0 |
| 960 | 0 | 1 | 0 | 1 | 0 |
| 961 | 0 | 0 | 0 | 0 | 0 |
| 962 | 0 | 0 | 0 | 0 | 0 |
| 963 | 0 | 0 | 1 | 1 | 1 |
| 964 | 0 | 0 | 1 | 0 | 0 |
| 965 | 0 | 0 | 1 | 0 | 0 |
| 966 | 0 | 0 | 0 | 0 | 0 |
| 967 | 0 | 0 | 1 | 1 | 1 |
| 968 | 1 | 0 | 0 | 1 | 0 |
| 969 | 0 | 0 | 0 | 0 | 0 |
| 970 | 0 | 0 | 0 | 1 | 0 |
| 971 | 0 | 0 | 0 | 0 | 0 |
| 972 | 0 | 0 | 0 | 0 | 0 |
| 973 | 0 | 0 | 0 | 0 | 0 |
| 974 | 0 | 1 | 0 | 0 | 1 |
| 975 | 0 | 0 | 0 | 1 | 0 |
| 976 | 0 | 0 | 0 | 1 | 0 |
| 977 | 1 | 0 | 0 | 1 | 0 |
| 978 | 0 | 0 | 0 | 0 | 0 |
| 979 | 1 | 0 | 1 | 1 | 1 |
| 980 | 1 | 0 | 0 | 1 | 0 |
| 981 | 0 | 0 | 0 | 1 | 0 |
| 982 | 0 | 0 | 0 | 1 | 0 |
| 983 | 0 | 0 | 0 | 0 | 0 |
| 984 | 0 | 0 | 0 | 0 | 0 |
| 985 | 0 | 0 | 0 | 0 | 0 |
| 986 | 1 | 1 | 1 | 0 | 1 |
| 987 | 0 | 0 | 0 | 0 | 0 |
| 988 | 0 | 0 | 0 | 0 | 0 |
| 989 | 0 | 0 | 0 | 1 | 0 |
| 990 | 0 | 0 | 0 | 0 | 0 |
| 991 | 1 | 1 | 0 | 1 | 0 |
| 992 | 0 | 0 | 0 | 0 | 0 |
| 993 | 0 | 1 | 0 | 0 | 1 |
| 994 | 1 | 1 | 0 | 1 | 0 |
| 995 | 0 | 0 | 0 | 1 | 0 |
| 996 | 0 | 0 | 1 | 0 | 0 |
| 997 | 0 | 0 | 0 | 1 | 0 |
| 998 | 1 | 0 | 1 | 1 | 1 |
| 999 | 0 | 0 | 0 | 0 | 0 |
| 1000 | 1 | 0 | 0 | 1 | 0 |
True causal effect (based on potential outcomes)
Conditional effect
The true conditional causal effect of \(A\) is 1.
mc<-glm(Y ~ X + C , data = dtB, family="binomial")
mc
Call: glm(formula = Y ~ X + C, family = "binomial", data = dtB)
Coefficients:
(Intercept) X C
-2.74 1.21 1.61
Degrees of Freedom: 999 Total (i.e. Null); 997 Residual
Null Deviance: 964
Residual Deviance: 797 AIC: 803
This estimate for \(X\) is a good estimate of the conditional effect in the population, based on the potential outcomes at each level of \(C\)
Marginal effect
The marginal estimate is biased both for the conditional effect and the marginal causal effect.
mm<-glm(Y ~ X , data = dtB, family="binomial")
mm
Call: glm(formula = Y ~ X, family = "binomial", data = dtB)
Coefficients:
(Intercept) X
-2.33 1.59
Degrees of Freedom: 999 Total (i.e. Null); 998 Residual
Null Deviance: 964
Residual Deviance: 876 AIC: 880
Numerator and Denominator model
Stabilized weights by hand
dtB[, pX0 := predict(numModel, type = "response")]
dtB[, pX := predict(denModel, type = "response")]
defB2 <- defDataAdd(varname = "IPW",
formula = "(X*pX0+(1-X)*(1-pX0))/((X * pX) + ((1 - X) * (1 - pX)))",
dist = "nonrandom")
dtB <- addColumns(defB2, dtB)
dtB[1:6]| id | C | Y0 | Y1 | X | Y | pX0 | pX | IPW |
|---|---|---|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 0 | 0.423 | 0.3347 | 0.8673 |
| 2 | 0 | 0 | 1 | 0 | 0 | 0.423 | 0.3347 | 0.8673 |
| 3 | 1 | 0 | 0 | 1 | 0 | 0.423 | 0.6718 | 0.6297 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0.423 | 0.3347 | 0.8673 |
| 5 | 1 | 1 | 1 | 0 | 1 | 0.423 | 0.6718 | 1.7578 |
| 6 | 1 | 0 | 0 | 1 | 0 | 0.423 | 0.6718 | 0.6297 |
unique(dtB$IPW)[1] 0.8673 0.6297 1.7578 1.2639
Stabilized weights with ipw package
tempsw<-ipw::ipwpoint(exposure=X,family="binomial",link="logit",numerator=~1,denominator = ~C,data=dtB)
tempw<-ipw::ipwpoint(exposure=X,family="binomial",link="logit",denominator = ~C,data=dtB)
unique(tempsw$ipw.weights)[1] 0.8673 0.6297 1.7578 1.2639
mean(tempsw$ipw.weights)[1] 1
unique(tempw$ipw.weights)[1] 1.503 1.489 3.047 2.988
mean(tempw$ipw.weights)[1] 2
Applying IPW
Unstabilized weights
Stabilized weights
Comparison
modelsummary::modelsummary(models=list("marginal"=mm,"conditional"=mc,"IPW"=mw,"IPWs"=mws))Warning in eval(family$initialize): non-integer #successes in a binomial glm!
Warning in eval(family$initialize): non-integer #successes in a binomial glm!
| marginal | conditional | IPW | IPWs | |
|---|---|---|---|---|
| (Intercept) | -2.333 | -2.736 | -2.069 | -2.069 |
| (0.147) | (0.165) | (0.100) | (0.132) | |
| X | 1.587 | 1.211 | 1.081 | 1.081 |
| (0.180) | (0.191) | (0.123) | (0.171) | |
| C | 1.614 | |||
| (0.183) | ||||
| Num.Obs. | 1000 | 1000 | 1000 | 1000 |
| AIC | 880.1 | 802.5 | 1847.3 | 1004.4 |
| BIC | 889.9 | 817.3 | 1857.1 | 1014.2 |
| Log.Lik. | -438.049 | -398.272 | -921.632 | -500.206 |
| F | 77.828 | 71.500 | 77.460 | 39.837 |
| RMSE | 0.37 | 0.35 | 0.37 | 0.37 |
We used R version 4.4.1 (R Core Team 2024) and the following R packages: ipw v. 1.2.1 (van der Wal and Geskus 2011), Matrix v. 1.7.0 (Bates, Maechler, and Jagan 2024), rio v. 1.0.1 (Chan et al. 2023), simstudy v. 0.7.1 (Goldfeld and Wujciak-Jens 2020), survey v. 4.4.2 (Lumley 2004, 2010, 2024), survival v. 3.6.4 [@].